$12^{3}_{37}$ - Minimal pinning sets
Pinning sets for 12^3_37
Minimal pinning semi-lattice
(y-axis: cardinality)
Pinning semi lattice for 12^3_37
Pinning data
Pinning number of this multiloop: 5
Total number of pinning sets: 128
of which optimal: 1
of which minimal: 1
The mean region-degree (mean-degree) of a pinning set is
on average over all pinning sets: 2.90623
on average over minimal pinning sets: 2.0
on average over optimal pinning sets: 2.0
Refined data for the minimal pinning sets
Pin label
Pin color
Regions
Cardinality
Degree sequence
Mean-degree
A (optimal)
•
{1, 2, 4, 6, 9}
5
[2, 2, 2, 2, 2]
2.00
Data for pinning sets in each cardinal
Cardinality
Optimal pinning sets
Minimal suboptimal pinning sets
Nonminimal pinning sets
Averaged mean-degree
5
1
0
0
2.0
6
0
0
7
2.38
7
0
0
21
2.65
8
0
0
35
2.86
9
0
0
35
3.02
10
0
0
21
3.14
11
0
0
7
3.25
12
0
0
1
3.33
Total
1
0
127
Other information about this multiloop
Properties
Region degree sequence: [2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 6, 6]
Minimal region degree: 2
Is multisimple: No
Combinatorial encoding data
Plantri embedding: [[1,2,3,3],[0,3,4,4],[0,5,5,6],[0,6,1,0],[1,7,8,1],[2,8,9,2],[2,9,9,3],[4,9,8,8],[4,7,7,5],[5,7,6,6]]
PD code (use to draw this multiloop with SnapPy): [[8,16,1,9],[9,15,10,14],[7,20,8,17],[15,1,16,2],[10,13,11,14],[17,6,18,7],[19,2,20,3],[4,12,5,13],[11,5,12,6],[18,4,19,3]]
Permutation representation (action on half-edges):
Vertex permutation $\sigma=$ (9,8,-10,-1)(1,12,-2,-9)(20,3,-13,-4)(11,4,-12,-5)(16,7,-17,-8)(2,13,-3,-14)(19,14,-20,-15)(15,18,-16,-19)(6,17,-7,-18)(5,10,-6,-11)
Edge permutation $\epsilon=$ (-1,1)(-2,2)(-3,3)(-4,4)(-5,5)(-6,6)(-7,7)(-8,8)(-9,9)(-10,10)(-11,11)(-12,12)(-13,13)(-14,14)(-15,15)(-16,16)(-17,17)(-18,18)(-19,19)(-20,20)
Face permutation $\varphi=(\sigma\epsilon)^{-1}=$ (-1,-9)(-2,-14,19,-16,-8,9)(-3,20,14)(-4,11,-6,-18,15,-20)(-5,-11)(-7,16,18)(-10,5,-12,1)(-13,2,12,4)(-15,-19)(-17,6,10,8)(3,13)(7,17)
Multiloop annotated with half-edges
12^3_37 annotated with half-edges